Protecting the Blues in Linanthus Parryae

Evolution, 55(7), 2001, pp. 1283–1298

STABLE TWO-ALLELE POLYMORPHISMS MAINTAINED BY FLUCTUATING FITNESSES AND SEED BANKS: PROTECTING THE BLUES IN LINANTHUS PARRYAE

MICHAEL TURELLI,1 DOUGLAS W. SCHEMSKE,2,3 AND PAULETTE BIERZYCHUDEK4 1Section of Evolution and Ecology and Center for Population Biology, University of California, One Shields Avenue, Davis, California 95616 E-mail: [email protected] 2Department of Botany, Box 355325, University of Washington, Seattle, Washington 98195-5325 E-mail: [email protected] 4Department of Biology, Lewis and Clark College, Portland, Oregon 97219-7899 E-mail: [email protected]

Abstract. Motivated by data demonstrating fluctuating relative and absolute fitnesses for white- versus blue-flowered morphs of the desert annual Linanthus parryae, we present conditions under which temporally fluctuating selection and fluctuating contributions to a persistent seed bank will maintain a stable single-locus polymorphism. In L. parryae, blue flower color is determined by a single dominant allele. To disentangle the underlying diversity-maintaining mechanism from the mathematical complications associated with departures from Hardy-Weinberg genotype frequencies and dominance, we successively analyze a haploid model, a diploid model with three distinguishable genotypes, and a diploid model with complete dominance. For each model, we present conditions for the maintenance of a stable polymorphism, then use a diffusion approximation to describe the long-term fluctuations associated with these polymorphisms. Our protected polymorphism analyses show that a genotype whose arithmetic and geometric mean relative fitnesses are both less than one can persist if its relative fitness exceeds one in years that produce the most offspring. This condition is met by data from a population of L. parryae whose white morph has higher fitness (seed set) only in years of relatively heavy rain fall. The data suggest that the observed polymorphism may be explained by fluctuating selection. However, the yearly variation in flower color frequencies cannot be fully explained by our simple models, which ignore age structure and possible selection in the seed bank. We address two additional questions—one mathematical, the other biological—concerning the applicability of diffusion approximations to intense selection and the applicability of long-term predictions to datasets spanning decades for populations with long-lived seed banks.

Key words. Diffusion approximation, ﬂower color polymorphism, ﬂuctuating selection, Frank’s distribution, seed bank.

Received July 21, 2000. Accepted October 23, 2000.

Through the seminal papers of Epling and Dobzhansky long run the product of selection operating at an intensity we (1942) and Wright (1943a, b), the flower color polymorphism have been unable to measure.’’ in the diminutive desert annual Linanthus parryae played a Schemske and Bierzychudek (2001) report more than a central role in shaping population genetic theory and opinions decade of fitness data that strongly support Epling et al.’s concerning the roles of natural selection, genetic drift, and (1960) interpretation. Only a cursory summary of the data migration in evolution. In their 1941 survey of 427 sampling will be provided here to motivate our mathematical analyses. locations in the Mojave Desert, California, Epling and Dob- At two sites with monomorphic populations, Schemske and zhansky (1942) found a preponderance of monomorphic pop- Bierzychudek (2001) performed transplant experiments that ulations with most having white flowers. About 19% of their showed directional selection favoring the local morph. In an 1261 samples were polymorphic, and white flowers predom- area that was polymorphic in 1941 and has remained poly- inated in most of these. Wright (1943b; 1978, pp. 194–223) morphic in recent surveys, they found fluctuating directional analyzed the spatial pattern of flower color frequencies and selection in seed production, with white-flowered plants out concluded that despite some evidence for directional selec- producing blue-flowered plants in wetter-than-average years tion favoring white, selection was generally negligible, and and blue-flowered plants out producing whites in dryer-than- drift, migration and, possibly, mutation-selection balance de- average years. In addition to the fluctuating relative fitnesses, termined allele frequencies in these populations. Wright ar- they also found dramatic fluctuations in plant density and gued that the effects of drift could explain the spatial pattern absolute seed production. In very dry years, no plants ap- of polymorphic frequencies if effective local population sizes peared; in dry years, there were five to 30 seeds per plant; were on the order of 100 or less. However, after monitoring while in years with high rainfall, individual plants produced these same populations for another 15 years, Epling et al. hundreds of seeds. Simultaneously, plant densities varied (1960) found little change in local flower-color frequencies. more than 40-fold among years in which seeds were pro- Based on this constancy and experimental evidence for a duced. Schemske and Bierzychudek’s data show that selec- long-lived seed bank, Epling et al. (1960) concluded that the tion contributes to the spatial pattern of flower-color fre- importance of genetic drift had been overestimated and that quencies. The question we address here is whether the ob- ‘‘the frequencies of blue and white flowered plants are in the served temporal variation in absolute and relative fitnesses is sufficient to explain persistent local polymorphism without 3 Present address: Department of Botany and Plant Pathology, invoking migration between habitats in which selection fa- Michigan State University, East Lansing, Michigan 48824-1312. vors the alternative morphs. The analysis is greatly simplified 1283 q 2001 The Society for the Study of Evolution. All rights reserved. 1284 MICHAEL TURELLI ET AL. by the fact that the Linanthus flower color dimorphism is quantitative predictions in terms of estimable parameters controlled by a diallelic locus with blue dominant to white rather than capturing all aspects of the biology. We first de- (Epling et al. 1960; Wright 1978, pp. 194–195; P. Bierzy- velop a sequence of simple models that illustrate the key chudek, unpubl. data). ideas. We compare exact conditions for the maintenance of Before addressing the data, we will present a mathematical stable polymorphisms with simpler conditions derived from analysis of conditions under which temporally fluctuating diffusion approximations. We use the diffusions to charac- selection combined with fluctuating contributions to a seed terize the polymorphisms and test the accuracy of their pre- bank can maintain variation at a diallelic locus. Dempster dictions for the mean and standard deviation of allele fre- (1955) provided the first correct analysis of temporally fluc- quencies against numerical simulations. Finally, we apply the tuating selection (reviewed in Gillespie 1991, sec. 4.7). He theoretical results to some L. parryae data from Schemske analyzed a haploid model and demonstrated that fluctuating and Bierzychudek (2001). Readers uninterested in the math- selection led to the fixation of the genotype with the highest ematical details should skip to the section analyzing the Lin- geometric mean fitness. Haldane and Jayakar (1963) showed anthus data after reading the introductory descriptions of the that fluctuating selection on diploids could maintain poly- models. morphisms. Assuming discrete generations and complete dominance, they found that a stable polymorphism would be MODELS AND ANALYSES maintained if the relative fitness of the recessive genotype, Model denoted w, satisfied two conditions: Its arithmetic mean, E(w), must exceed one; but its geometric mean, Throughout we will consider a single, diallelic autosomal exp{E[ln(w)]}, must be less than one. Gillespie has developed locus in an annual, with alleles denoted A1 and A2.Thelong- the most extensive body of theory delimiting conditions un- term fate of these alleles will be determined by their persis- der which fluctuating selection will maintain genetic variation tence in the seed bank. For simplicity, we will assume that (see Gillespie 1991), but he has not specifically addressed selection acts only on adult fecundity. Epling et al. (1960) the effects of overlapping generations. and Wright (1978, pp. 194–223) report that although ger- Templeton and Levin (1979) analyzed temporally fluctu- mination rates vary dramatically across years, genotype-spe- ating selection on plants with seed banks. They used nu- cific differences in germination rates seemed negligible. To merical simulations of complex models involving age struc- study the properties of the seed bank, they removed flowering plants before seed-set for several years in experimental plots. ture and age-dependent selection both during dormancy and As estimated by Wright (1978, pp. 196–197), seedling den- after germination to demonstrate that seed banks facilitate sity declined by only about 50% after 7 years, indicating a the maintenance of polymorphism. Subsequently, consider- long-lived seed bank. Although Wright (1978, p. 197) con- able analytical progress has been made in understanding sto- cluded that there was no apparent viability selection among chastic age-structured models (e.g., Tuljapurkar 1990; Or- the dormant seeds, our x2 test of homogeneity from the three zack 1993). However, we cannot apply these models because site B samples reported in figure 10 of Epling et al. (1960) the necessary age-specific data on seed survival and germi- shows statistically significant differences among years in nation are not available. The simple models we analyze below flower-color frequencies, despite the fact that the samples in are much closer to the ecological lottery-competition models the first and last years are not significantly different. Thus, investigated by Chesson and his collaborators (e.g., Chesson Wright’s conclusion of no consistent differential viability and Warner 1981; Hatfield and Chesson 1989; Chesson seems valid, but the data also suggest genotype- and/or age- 1994). They have shown that coexistence of competing spe- specific year-to-year variation in germination and/or seedling cies is facilitated by the combined effects of overlapping survival rates. Given that we have no data on these com- generations and fluctuating conditions that favor different ponents of fitness, we ignore them in our mathematical anal- competitors in different breeding seasons. Similar analyses yses, but reconsider them below. have been adapted to the maintenance of genetic variation Given the available data, we will not partition male and for multiple alleles and loci by Ellner and his collaborators female reproductive functions and will assume that the key (e.g., Ellner and Hairston 1994; Ellner and Sasaki 1996; Ell- fitness differences can be captured by estimating the average ner et al. 1999). Seger and Brockmann (1987) and Hedrick number of seeds produced by each phenotype. We assume (1995) demonstrated the polymorphism-preserving role of that viability of seeds and germination rates are independent seed banks with simple one-locus, two-allele models that of genotype at this locus and that all seeds in the seed bank ignore season-to-season variation in seed set. Our slightly are equivalent, independent of age. These simplifying as- more complex models, which include variation in productiv- sumptions are not likely to be valid in the field. Our hope is ity, analytically capture a key insight of Templeton and Lev- that the effects of the large observed differences in absolute in’s (1979) simulations—the importance of covariance be- and relative seed set dominate the smaller effects associated tween seed production and relative fitnesses. Our attempt at with fitness components that have not been estimated. These understanding the L. parryae polymorphism in terms of a assumptions are testable, as demonstrated below, and our specific model complements Subramaniam and Rausher’s model can be generalized toward Templeton and Levin’s (2000) experimental approach to demonstrating that balanc- (1979) more realistic treatment as additional data become ing selection maintains the flower-color polymorphism in the available. morning glory, Ipomoea purpurea. Let pt denote the frequency of allele A1 in the seed bank Our goal is to make empirically useful qualitative and just before germination in year t. Because there is no effect FLOWER DIMORPHISM AND FLUCTUATING SELECTION 1285 of genotype on germination, pt is also the allele frequency of such models and their diffusion approximations, see Tur- among new seedlings. Letpt9 denote the frequency of A1 elli 1977). Essentially the same condition was derived by among seeds produced in year t. We assume that just before Chesson and Warner (1981), who demonstrated the potential germination in year t 1 1, a fraction at of all seeds in the for environmental variation to promote the coexistence of seed bank was produced by adult plants in year t. Thus, the competing species (compare eqs. 3 and 12 of Chesson and recursion for pt is Warner 1981). The case with at constant was discussed by Seger and Brockmann (1987). Because ln(1 1 x)isaconcave p 5 (1 2 a )p 1 ap9. (1) t11 tt tt function (i.e., has a negative second derivative), Jensen’s Favorable conditions will be associated with relatively large inequality implies that values of at, corresponding to greater input into the seed bank. ww1,t 1,t Our frequency-based model ignores the ecological complex- E ln 1 1 att2 1 # ln 1 1 E a 2 1 , (5) ities that govern the absolute size of the seed bank. []12ww2,t 5 []122,t 6 Analyzing the diploid model relevant to Linanthus is com- where equality holds only if there is no variation. It follows plicated by dominance and the fact that the seed bank consists from inequalities (4) and (5) that a necessary condition for of a mixture of genotypes produced by selection in different the protection of A1 is generations. The mixture produces a temporal Wahlund effect w1,t (Christiansen 1988), so that the seedling genotype frequen- E at 2 1 . 0. (6) cies will generally not follow Hardy-Weinberg proportions, []12w2,t and the allele frequency, pt, is insufficient to describe the To clarify the role of the fluctuating contributions to the exact evolutionary dynamics. Because these mathematical seed bank, let at 5 azt, where a 5 E(at)measurestheaverage complications obscure the forces maintaining variation, we fraction of each generation’s pregermination seed bank that analyze in turn three different genetic models: (1) haploid consists of seeds produced the previous year. We assume that selection; (2) diploid selection with heterozygotes having in- 0 , a , 1. The random variable zt describes the fluctuations termediate fitness; and (3) diploid selection with complete in seed production, by definition it satisfies E(zt) 5 1. With dominance. We focus on deriving analytical conditions for this notation, (6) reduces to the maintenance of stable polymorphisms, then present diffusion approximations to describe the properties of these ww1,t 1,t w1,t E ztt5 E 1 Cov z , . 1. (7) polymorphisms. Our general conclusions relate to the main- 1212ww2,t 2,t 1 w2,t 2 tenance of single-locus polymorphisms in any population Condition (7) requires that the weighted relative fitness of with discrete breeding seasons and overlapping generations. A1 exceed one, where the weightings involve the fluctuating contributions to the seed bank. The weighting implies that Selection on Haploids A1 can persist even if it performs relatively poorly on average For haploids (or diploids with complete selfing), the evo- (so that E(w1,t/w2,t) , 1), as long as this is offset by per- forming sufficiently better than A in years that contribute lutionary dynamics are completely characterized by pt. Let- 2 most to the seed bank (so that Cov[z ,w /w ]issufficiently ting w1,t (w2,t)denotethefitnessofgenotypeA1 (A2)ingen- t 1,t 2,t eration t, we have large). An empirical example is provided by the L. parryae data discussed below. pw t 1,t The condition analogous to (4) for the protection of A2 pt95 (2) pwt 1,tt1 (1 2 p )w2,t (i.e., P[pt → 1ast → `] 5 0) is in equation (1). We seek conditions for stochastic protected w2,t polymorphism (Prout 1968), that is, conditions that preclude E ln 1 1 at 2 1 . 0. (8) []12w1,t either allele frequency from converging to zero over time. We will not provide a rigorous mathematical analysis; how- As above, a necessary condition for A2 to persist is ever, it can be shown (cf. Karlin and Liberman 1975) that ww2,t 2,t w2,t the following heuristic argument produces the correct poly- E ztt5 E 1 Cov z , . 1. (9) 1212ww1,t 1,t 1 w1,t 2 morphism conditions. When A1 is extremely rare, that is, pt ˘ 0, the dynamics of recursion (1) can be approximated by When both conditions (4) and (8) are met, we have a protected polymorphism; it is necessary, but not sufficient, to have both ww1,t 1,t (7) and (9) satisfied. pt11 5 (1 2 att)p 1 ap tt5 p t1 2 a t1 a t . (3) ww2,t 122,t As a approaches zero, conditions (7) and (9) do suffice for the maintenance of protected polymorphism (Chesson and This corresponds to the standard recursion for population Warner 1981). When there is no correlation between a and growth in a random environment, so the condition for p to t t the relative fitnesses, for instance when a is constant, these stay away from zero in the long run (i.e., P[p → 0ast → t t necessary conditions reduce to `] 5 0) is simply w1,t w1,t E . 1 and (10a) E ln 1 1 at 2 1 . 0, (4) 12w2,t []12w2,t w where E denotes mathematical expectation over the distri- E 2,t . 1. (10b) bution of the random vector (at,w1,t, w2,t)(foradiscussion 12w1,t 1286 MICHAEL TURELLI ET AL.

As illustrated below, these conditions, which may seem mu- the accuracy of these predictions, so numerical simulations tually exclusive, are both satisfied whenever the differences are needed to evaluate their usefulness. For simplicity, we between the arithmetic means of the two relative fitnesses restrict our analyses by assuming that the random variables are small relative to their variances. (at,w1,t, w2,t)areindependentfromgenerationtogeneration, With no seed bank, that is, with at [ 1, no protected poly- so that (1) is a Markov process (cf. Gillespie 1991, ch. 4). morphism is possible under this model. As shown by Demps- Wright (1948) pioneered diffusion approximations of fluc- ter (1955), the population becomes monomorphic for the al- tuating selection. However, mathematical errors led both lele with the higher geometric mean fitness, that is, the higher Wright (1948) and Kimura (1954) to conclude mistakenly value of E[ln(wi,t)]. As a 5 E(at)decreases,therangeof that fluctuating selection could not maintain genetic varia- parameters producing a protected polymorphism increases to tion. Their analyses were corrected by Gillespie (1973, 1974) the region defined by (10a, b). To quantify this effect simply, and extended to diploids by Levikson and Karlin (1975; for we will assume that the fitnesses follow lognormal distri- treatment of an ecological analog of eq. 1, see Hatfield and butions, so that ln(w1,t)andln(w2,t)follownormaldistribu- Chesson 1989). tions with means m and m ,variancesss22 and ,andcor- 1 2 12 Diffusion approximations assume that selection is rela- relation r.Thevariabilityoftherelativefitnessescanbe tively weak and the stochastic fluctuations in the parameters expressed in terms of s2 5 Var[ln(w /w )] 512ss22 1,t 2,t 12 are relatively small (see the Appendix for an elaboration). 2rs s .Thedifferenceinthegeometricmeanfitnessesof 1 2 We present a new approach for applying these weak-selec- the genotypes can be quantified by zm 2mz 5 zE[ln(w / 1 2 1,t tion, small-noise approximations to biological circumstances w2,t)]z.Thelognormalassumptionsimplythatw1,t/w2,t also 2 in which selection is strong and stochastic fluctuations ex- has a lognormal distribution with mean exp[m1 2m2 1 (s / 2)] and coefficient of variation (CV)œexp(s2) 2 1 . Thus, treme then explore the range of parameter values for which the necessary conditions (10a, b) are satisfied whenever the approximations are informative. The diffusion approximation characterizes the model’s behavior by the infinitesi- Var[ln(w1,t/w2,t)] . 2zE[ln(w1,t/w2,t)]z, (11) mal mean, denoted m(p), and the infinitesimal variance, de- which implies that differences in the geometric means must noted v(p), which approximate E(Dpt z pt 5 p)andVar(Dpt z pt be compensated by sufficient fluctuation in the relative fit- 5 p), respectively (Karlin and Taylor 1981, ch. 15, especially nesses (cf. Chesson and Warner 1981, fig. 1). The expected pp. 184–188). The Appendix shows that values in conditions (4) and (8) can be obtained numerically using Mathematica (Wolfram 1996). These exact results, 1 m(p) 5 ap(1 2 p) g1C 1s2 2p and (13a) which require specifying the full distributions of all of the []122 random variables, will be numerically compared below to the 22 2 results of a diffusion approximation, which depends only on v(p) 5 a s [ p(1 2 p)] , (13b) means, variances and covariances. where g5m2Ωs2, m denotes the average (arithmetic mean) In general, superimposing fluctuations in at that are in- 2 dependent of the relative fitnesses will make the protected intensity of selection in favor of A1, s denotes the variance polymorphism conditions slightly more restrictive. To see in relative fitness, and C denotes the covariance between zt 2 this, use the Jensen’s inequality argument from (5), after and the relative fitness of A1.Notethatsz 5 Var(zt) 5 2 conditioning on the values of w1,t and w2,t,toobtain [CV(at)] does not enter these expressions, reflecting the fact that to this order of approximation, variation in zt contributes ww1,t 1,t E ln 1 1 att2 1 , E ln 1 1 E(a ) 2 1 , (12) to the allele frequency evolution only when it covaries with [][]12ww2,t 122,t the relative fitnesses (i.e., through the term C in eq. 13a). whenever at varies. This small effect will be quantified below Gillespie (1991, ch. 4) discusses this same diffusion as his along with the much larger effects of fluctuations in at that ‘‘c-haploid model,’’ including the consequences of autocor- are correlated with w1,t/w2,t. relation. When diffusion approximations are applied to discrete- Diffusion approximation and numerical results time models in which the weak-selection, small-noise assumptions are not satisfied, there is a fundamental ambiguity The analyses above provide conditions for protected polymorphism. But for many purposes, we will want to know in identifying the diffusion parameters with the moments of properties of these polymorphisms, such as the long-term the random variables from the discrete-time model. For sim- mean and variance of the allele frequencies. These are very ple selection models like (1), we propose resolving this am- difficult to obtain analytically for the discrete-time, nonlinear biguity with the following consistency criterion: the param- stochastic process defined by (1) and (2), but they can be eters must be chosen so that the resulting conditions for pro- approximated using diffusion theory. This process involves tected polymorphism and the moments of the allele frequency approximating a discrete-time process by a continuous-time distribution are independent of whether we quantify selection process and is justified by assuming weak selection. As dis- by examining the relative fitnesses w1,t/w2,t or w2,t/w1,t.Be- cussed below, this requires approximating the stochastic se- cause this arbitrary choice for normalizing fitnesses does not lection process and leads to conditions for polymorphism that affect the actual dynamics, a consistent diffusion approxi- generally differ somewhat from the exact conditions derived mation must be invariant to these alternative normalizations. from the original model. There is no general theory to bound This consistency criterion leads to FLOWER DIMORPHISM AND FLUCTUATING SELECTION 1287

w g5E ln1,t , (14a) []12w2,t w s52 Var ln1,t , and (14b) []12w2,t

w1,t C 5 Cov zt, ln (14c) []12w2,t (see the Appendix for details). The diffusion process corresponding to (13) possesses a nontrivial, stationary distribution, corresponding to protected polymorphism, if and only if (1 2 a)s2 zg1Cz , (15) 2 (Karlin and Taylor 1981, pp. 220–221). When g1C . Ω(1 2 2 a)s ,A1 spreads to fixation; whereas when g1C ,2Ω(1 2 2 a)s ,A2 spreads to fixation. Note that when at [ 1forall t, corresponding to haploid selection with no seed bank, the protected polymorphism condition (15) cannot be satisfied. FIG.1. Conditionsforhaploidprotectedpolymorphismswithlog- In this case, the approximation exactly reproduces Demps- normally distributed fitnesses. The symbols indicate numerical ap- ter’s (1955) conclusion from the discrete-time model: A1 proximations for the minimum variance in log relative fitnesses, spreads to fixation with probability one if and only if its Var[ln(w2,t/w1,t)], necessary to produce a protected polymorphism despite a geometric mean fitness disadvantage for allele A ,as geometric mean fitness exceeds that of A2,thatis,g5 1 quantified by E[ln(w2,t/w1,t)] . 0. The lines are interpolations of E[ln(w1,t/w2,t)] . 0. the numerical values, except for the line corresponding to a → 0, Condition (15) makes several predictions. First note that which is given by (11). The predictions from (15) are always within if C 5 0, implying that variation in contributions to the seed 16% of the numerically derived values, they remain within 9% for bank is uncorrelated with variation in relative fitnesses (i.e., zE[ln(w1,t/w2,t)]z # 0.5. See the text for additional details. neither allele is systematically favored in good years), variation in zt does not affect the conditions for protected polymorphism. Second, note that g and C enter (15) only as a a decreases, corresponding to a longer-lived seed bank, less sum. Thus, if allele A1 has higher relative fitness in good variation in the relative fitnesses is needed to compensate for years, this effectively raises its geometric mean relative fit- agivengeometricmeanfitnessdisadvantage.Theratioof ness from g5E[ln(w1,t/w2,t)] to E[ln(w1,t/w2,t)] 1 Cov[zt, the slopes of the best-fit linear regressions (forced through ln(w1,t/w2,t)]. Third, for a , 1, condition (15) predicts that the origin) to the numerically obtained minimum values of the minimum variance necessary to produce a protected poly- Var[ln(w1,t/w2,t)] with a 5 0.1 versus a 5 0.5 is 1.91, whereas morphism increases linearly with zg1Cz.Fourth,(15)pre- (15) predicts a ratio of 1.8. Once a is as low as 0.1, the dicts that this minimum variance is proportional to 1/(1 2 minimum variance is closely approximated by the necessary a). condition (11), corresponding to a → 0. Figure 1 examines the third and fourth predictions. This The effect of varying at was quantified by assuming that figure compares the approximate protected polymorphism zt has a log-normal distribution truncated above at 1/a (so condition (15) to numerical results obtained from the exact that 0 , at , 1), whose moments were chosen so that E(zt) conditions (4) and (8) assuming that at is constant and w1,t/ 5 1andCV(at)wasequaltoaspecifiedvalue.Theminimum w2,t follows a lognormal distribution. The lines in Figure 1, value of Var[ln(w1,t/w2,t)] consistent with maintaining a pro- corresponding to constant values of a, provide the minimum tected polymorphism was determined numerically from (4) values of Var[ln(w1,t/w2,t)] necessary to achieve a protected and (8). For CV(at) # 0.3, the effect of variation in at is polymorphism for specified values of zE[ln(w1,t/w2,t)]z (cf. fig. generally too small to display on the scale of Figure 1. Using 1ofChessonandWarner1981;fig.1ofSegerandBrock- a 5 0.25 with 0.1 # E[ln(w2,t/w1,t)] # 1, the relative increase mann 1987; and fig. 1 of Hatfield and Chesson 1989). The in the critical value of Var[ln(w1,t/w2,t)] was less than 1% for results support the prediction that the minimum Var[ln(w1,t/ CV(at) 5 0.1 versus CV(at) 5 0, less than 2% for CV(at) 5 w2,t)] increases approximately linearly with zE[ln(w1,t/w2,t)]z. 0.2, and less than 3% for CV(at) 5 0.3. To appreciate the range of parameters explored, note that Figure 2 examines the predicted effects of Cov[zt, ln(w1,t/ when E[ln(w1,t/w2,t)] 5 1.0 (or 0.5), the geometric mean fit- w2,t)]. As expected, variation in at has a much larger effect ness of A1 (i.e., exp{E[ln(w1,t)]}) exceeds that of A2 by a when it systematically favors one of the alleles, example, factor of 2.72 (or 1.65). Not surprisingly, extreme levels of when Cov[zt, ln(w1,t/w2,t)] . 0, so that A1 tends to have a variation are needed to compensate for such large fitness higher relative fitness in years that make greater contributions differences. It is surprising, however, that our consistent dif- to the seed bank. This positive covariance lowers the mini- fusion approximation remains so accurate for such extreme mum variance in relative fitnesses needed to protect A1 from levels of selection and variation. As predicted from (15), as loss when it has a geometric mean fitness disadvantage (i.e., 1288 MICHAEL TURELLI ET AL.

morphism-promoting role of variation in relative fitnesses. The expression for SD(p)showsthatevenwhengeneticvar- iation is maintained by extreme fluctuations in fitness, allele frequencies will not fluctuate wildly if a is small, corresponding to prolonged dormancy in the seed bank. The accuracy of the diffusion predictions concerning protected polymorphism was tested with replicated Monte Carlo simulations of the discrete-time model. As in our analysis of protected polymorphism, we assumed a log-normal distribution for w1,t/w2,t with relative fitnesses independent across generations. Sample outcomes are presented in Figures 3 and 4. One hundred replicate simulations were performed for each parameter combination. Each simulation began at the expected allele frequency predicted by (17). After 500 generations, means and SDs were calculated from the allele frequencies generated over the next 10,000 generations. The values reported in Figures 3 and 4 for the means and SDs are averages of the means and SDs obtained from 100 replicates. The two panels in Figure 3 compare the predictions from FIG.2. Conditionsforhaploidprotectedpolymorphismswithlog- (17) to Monte Carlo results. For the parameter values used, normally distributed fitnesses when the relative fitnesses are cor- the diffusion-based criterion (15) predicts a stable polymor- related with the inputs to the seed bank. As in Figure 1, the symbols phism for zE[ln(w1,t/w2,t)]z , 0.267; whereas the exact in- indicate numerical approximations for the minimum variance in log vasion analysis, (4) and (8), requires zE[ln(w1,t/w2,t)]z , relative fitnesses, Var[ln(w2,t/w1,t)], necessary to produce a protected polymorphism despite a geometric mean fitness disadvantage for 0.272. Given the close agreement of these bounds and the allele A1.Forallofthesecalculations,E(at) 5 0.25. r denotes the fact that the average allele frequency must increase from zero correlation between zt, which describes the fluctuating contributions to one between the lower and upper bounds, the agreement to the seed bank, and ln(w1,t/w2,t), the relative fitness of A1.Seethe between the predicted and observed means in Figure 3A may text for additional details. seem unremarkable. There is no comparable constraint, however, on the SDs displayed in Figure 3B, which show a similar E[ln(w1,t/w2,t)] , 0). This effect can be quantified by assum- level of agreement between the predicted and observed val- ing that (w1,t/w2,t)andzt have correlated log-normal distri- ues. The two panels in Figure 4 compare the values predicted 2 butions, with zt truncated above at 1/a. (Lognormality con- by (17) to Monte Carlo results as s 5 Var[ln(w1,t/w2,t)] strains the correlation between zt and ln[w1,t/w2,t], details will increases. As predicted, the average allele frequency con- be provided on request.) Figure 2 shows that even moderate verges toward 1/2. positive correlations can significantly lower the variance needed to compensate for a geometric mean fitness disad- Selection on Diploids: Incomplete Dominance vantage. When condition (15) is satisfied, the stationary distribu- The existence of a seed bank complicates the protected tion, f(p), which describes the long-term distribution of the polymorphism analysis, because the allele frequency pt no longer suffices to describe the composition of the seed bank. frequency of A1,is Even when adults mate randomly, so that the newly produced G(b121 b ) f(p) 5 pb1221(1 2 p)b 21 , (16a) seeds follow Hardy-Weinberg proportions, fluctuating allele G(b12)G(b ) frequencies with at , 1willgenerateadeparturefromHardy- with Weinberg proportions among seedlings (Christiansen 1988). Acompletedescriptionoftheevolutionarydynamicsrequires 2(g1C) 1 (1 2 a)s2 following genotype frequencies. Let f(AiAj)t denote the fre- b1 5 2 and (16b) as quency of seeds of genotype AiAj in the seed bank, just before germination in year t. We assume that the seed production 22(g1C) 1 (1 2 a)s2 of an adult is proportional to both its rate of pollen export b2 5 2 . (16c) as and ovule production. We also assume random mating. With (see Karlin and Taylor 1981, pp. 220–221; Turelli 1981). these assumptions, the combined effects of viability and fe- This is a beta distribution with mean and standard deviation cundity differences among genotypes can be summarized by viability-analogous fitnesses (Bodmer 1965), denoted w . 1 g1C ij,t E(p) 51 and (17a) The effective frequency of AiAj adults in generation t is 2 s2(1 2 a) a 2(g1C) 2 f(Aij A )w ij,t SD(p) 5 1 2 . (17b) f(Aijt A )95 , (18) 4(2 2 a)[]12s(1 2 a) w t 2 Note that E(p) → Ω as s increases, displaying the poly- with w t 5 f(A1A1)w11,t 1 f(A1A2)w12,t 1 f(A2A2)w22,t.The FLOWER DIMORPHISM AND FLUCTUATING SELECTION 1289

FIG.3. ComparisonofMonteCarloresults(dots)withthediffusionpredictions(solidcurve)forthemean(panelA)andthestandard 2 deviation (SD, panel B) of allele frequencies under haploid selection. The calculations assume that at [ 0.25 and s 5 Var[ln(w1,t/w2,t)] 5 0.3, whereas g5E[ln(w1,t/w2,t)] ranges from 20.11 to 0.11. Standard errors for the estimates were obtained from the 100 independent replicates, but were too small to display in the figures—the maximum value was 0.0028 for the estimates of E(p)and0.0025forthe estimates of SD(p). seeds produced by these adults follow Hardy-Weinberg pro- Although departures from Hardy-Weinberg are potentially portions with allele frequency significant for intermediate allele frequencies, the departures are very small when one of the alleles is extremely rare as 1 f(A ) [ p95f(A A )91 f(A A )9. (19) long as fitnesses do not fluctuate too severely (e.g., alternating 1 t 11t 2 12t generations of lethality for the alternative homozygotes). Un- der realistic levels of fitness variation, we can ignore the Thus, in the next generation, the genotype frequencies in the departures from Hardy-Weinberg when deriving the protected seed bank are given by polymorphism conditions, as demonstrated by our numerical analyses below. When either allele is rare, the genotype fre- f(A A ) 5 (1 2 a )f(A A ) 1 a (p9)2 and (20a) 11t11 t 11ttt quencies approximate Hardy-Weinberg and the protection f(A A ) 5 (1 2 a )f(A A ) 1 a 2p9 q9, (20b) conditions reduce to the haploid case (3) with the rare allele 12t11 t 12tttt being present only in heterozygotes. The resulting protected withpttt9 given by (19) and q9 5 1 2 p9. polymorphism conditions are

FIG.4. ComparisonofMonteCarloresults(dots)withthediffusionpredictions(solidcurve)forthemean(A)andSD(B)ofallele 2 frequencies under haploid selection. The calculations assume that at [ 0.25 and g5E[ln(w1,t/w2,t)] 520.1, whereas s 5 Var[ln(w1,t/ w2,t)] ranges from 0.27 to 5. The maximum standard errors for the estimates of the means (SDs) is 0.0037 (0.0049). 1290 MICHAEL TURELLI ET AL.

w12,t w2,t E ln 1 1 at 2 1 . 0 and (21a) E at 2 1 . 0, (26) []12w22,t []12w1,t which is the necessary condition derived above for the hap- w12,t E ln 1 1 at 2 1 . 0. (21b) loid case. This shows that complete recessivity facilitates the []12w11,t maintenance of an allele under this model. Condition (26) In preparation for analyzing the Linanthus polymorphism, also follows as a limit from (23), where the heterozygote has it is useful to consider the special case in which the hetero- intermediate fitness. Setting at 5 azt, as before, condition zygote always has intermediate fitness. For simplicity, we (26) can be rewritten as will assume that ww2,t 2,t w2,t E ztt5 E 1 Cov z , . 1. (27) w12,t 5 hw11,t 1 (1 2 h)w22,t (22) 1212ww1,t 1,t 1 w1,t 2 for some fixed dominance parameter h, with 0 , h , 1. With As in the haploid case, this shows that a positive covariance this simplification, the protected polymorphism conditions between favorable conditions (high zt)andtherelativefitness (21) become of A2 can compensate for a low arithmetic mean fitness. If at [ 1, corresponding to no seed bank, the protected poly- w11,t E ln 1 1 aht 2 1 . 0 and (23a) morphism conditions (25) and (27) reduce to the conditions []12w22,t derived by Haldane and Jayakar (1963).

w22,t E ln 1 1 at(1 2 h) 2 1 . 0. (23b) Diffusion approximation and numerical results []12w11,t As in the haploid case, we can again obtain a diffusion These are equivalent to the haploid conditions, (4) and (8), approximation, but now an additional approximation enters with at replaced by ath when considering protection of allele because of the departures from Hardy-Weinberg. We will A1 and at replaced by at(1 2 h)whenconsideringprotection apply a one-dimensional diffusion approximation, assuming of allele A2.AsshowninFigure1,loweringE(at)broadens that the genotype frequencies closely approximate Hardy- the range of parameter values for which polymorphism is Weinberg, then present simulation results showing that the maintained. Given that the dominance parameter effectively resulting predictions for the mean and SD of allele frequen- lowers E(at), conditions (23) show that diploidy with het- cies are quite accurate over a wide range of parameter values. erozygote intermediacy provides polymorphism-facilitating Proceeding as in the haploid case, the appropriate infini- buffering (as exemplified by Gillespie’s [1991, ch. 4.4] SAS- tesimal mean and variance are CFF model) that supplements the storage effect provided by 1 the seed bank (cf. Chesson 1985). As h → 1, corresponding m(p) 52ap(1 2 p)22g1 s 1C 2s 22(1 2 p) (28a) 2 to A1 becoming increasingly dominant, the sufficient con- [] dition for maintaining the nearly recessive allele A2 in the and population (i.e., inequality 23b) converges to the necessary condition v(p) 5 a2222[ p(1 2 p)]s , (28b) where w22,t E at 2 1 . 0. (24) 12w11,t w2,t [] g5E ln , (29a) The accuracy of these protected polymorphism conditions is []12w1,t discussed below. w s52 Var ln2,t , and (29b) Selection on Diploids: Dominance []12w1,t

Assume that allele A1 is dominant to A2,sothatw11,t 5 w C 5 Cov z , ln2,t (29c) w12,t 5 w1,t and w22,t 5 w2,t.Thecasewitha constant has t []12w1,t been analyzed numerically by Hedrick (1995), who assumed that genotype frequencies follow Hardy-Weinberg. From (note the change in sign between the definitions of g in equa- condition (23a), we see that the protected polymorphism con- tions 29a–c and 14). The behavior of this diffusion process is determined by a composite parameter, k, defined by dition for the dominant allele A1 is 1 w1,t g1 s2 1C E ln 1 1 at 2 1 . 0, (25) 2 []12w2,t k 5 . (30) s2 which is identical to the haploid condition (4). Deriving the protection condition for the recessive allele Astationarydistribution,correspondingtoprotectedpoly- is more subtle because a linear analysis does not suffice. The morphism, exists if and only if arguments of Haldane and Jayakar (1963) and Karlin and a Liberman (1975) can be adapted to show that, whenever the 0 , k , 1 2 , (31) 2 departures from Hardy-Weinberg are negligible for p ¯ 1, the recessive allele is protected whenever where the first inequality corresponds to protection of the FLOWER DIMORPHISM AND FLUCTUATING SELECTION 1291

FIG.5. ComparisonofMonteCarloresults(dots)withthediffusionpredictions(solidcurve)forthemean(A)andSD(B)ofallele 2 frequencies under diploid selection with complete dominance. The calculations assume that at [ 0.25 and s 5 Var[ln(w1,t/w2,t)] 5 0.3, whereas g5E[ln(w1,t/w2,t)] ranges from 20.10 to 0.148. Standard errors for the estimates were obtained from 100 replicates, but were too small to display in the figures—the maximum value was 0.0032 for the means and 0.0018 for the SDs. recessive allele and the second corresponds to protection of reduces to the Haldane and Jayakar (1963) model analyzed the dominant. The condition for protection of the dominant by Levikson and Karlin (1975) and the moments simplify to allele, k , 1 2 (a/2), can be rewritten as g1C , (s2/2)(1 2 a), which is identical to condition (15) derived for the 1 2 2k haploid model. When at [ 1, the condition for protection of E(p) 5 and (35a) the dominant reduces to Haldane and Jayakar’s (1963) con- 1 2 k dition, exp{E[ln(w2,t/w1,t)]} , 1. In other cases, conditions k2(1 2 2k) Var(p) 5 . (35b) (31) approximate the exact conditions (25) and (27). (1 2 k)2 When conditions (31) are satisfied, the stationary distribution for the frequency of A1 is described by Frank’s dis- Before using these results to discuss the Linanthus poly- tribution, morphism, we will compare the analytical predictions to Monte Carlo results produced by simulating the exact two- 2k 22k Exp Exp p2[(12k)/a21](1 2 p) (2k/a)21 dimensional recursions (18)–(20) for genotype frequencies. 12aa 1(1 2 p) 2 Sample outcomes are shown in Figures 5 and 6. The simu- f(p) 5 , lations followed the procedures used in the haploid examples 2(1 2 k) 2(1 2 k)2k 2k G21U 2 1, 4 2 , (Figs. 3 and 4). Despite our Hardy-Weinberg assumption, the 121aaaa 2one-dimensional diffusion approximation still accurately pre- (32) dicts the means and standard deviations. where U(x, y, z)denotesthesecondKummerconfluenthy- APPLICATION TO LINANTHUS pergeometric function (Abramowitz and Stegun 1964, eq. 13.2.6) and G(x)denotesthegammafunction.Itsmeanand Estimation of Parameters variance are Schemske and Bierzychudek (2001) have estimated plant E(p) 5 1 2 E(1 2 p) density, flower-color frequency, and seed set per plant at a site near Pearblossom, California (Pearblossom site 1) very 2(1 2 k)2k 2k U 2 1, 3 2 , close to sites surveyed in Epling and Dobzhansky (1942) and 12aaa Epling et al. (1960). Table 1 presents the data for the 7 years 5 1 2 and (33) between 1988 and 1999 with nonzero seed set. We will use 2(1 2 k)2k 2k U 2 1, 4 2 , these to estimate the parameters of our model. 12aaa First, we estimate relative fitnesses for each year from 2(1 2 k)2k 2k relative seed production. The results are reported in the fifth U 2 1, 2 2 , column of Table 1. Next, consider z 5 a /E(a ), the relative aaa t t t 12 year-to-year contributions to the seed bank. Let x Var(p) 52[E(1 2 p)]2 . (34) t denote the 2(1 2 k)2k 2k total number of seeds produced at the site in year t, and let U 2 1, 4 2 , x denote the average of the x . Using the Table 1 data, we 12aaa t estimate xt as the product of the number of plants found on When at [ 1, corresponding to no seed bank, the model three 50 3 1-m transects that provided most of the plants for 1292 MICHAEL TURELLI ET AL.

FIG.6. ComparisonofMonteCarloresults(dots)withthediffusionpredictions(solidcurve)forthemean(A)andSD(B)ofallele frequencies under diploid selection with complete dominance. The calculations assume that at [ 0.25 and g5E[ln(w1,t/w2,t)] 5 0.1, 2 whereas s 5 Var[ln(w1,t/w2,t)] ranges from 0.202 to 2. The maximum standard errors for the estimates of the means (SDs) is 0.0037 (0.0023). the seed set analysis and the average number of seeds per where the first condition insures persistence of white and the plant (weighted by the phenotype frequencies determined on second insures persistence of blue. Table 1 shows that al- the transects). We estimate zt in Table 1 as though E(wwhite/wblue) , 1, the critical weighted average, E(zt wwhite,t/wblue,t), exceeds one, reflecting the fact that whites xt zt 5 . (36) produce more seeds than blues in high rainfall years that x contribute disproportionately to the seed bank. As shown by According to our model, the average time that seeds remain (5)–(7) above, condition (38) for the persistence of blue can dormant is proportional to 1/a. Epling et al. (1960) reported be satisfied only if that plants reappeared on a plot on which no adults had been ww w seen in 10 years. They also performed removal experiments E z blue,t 5 E blue,t 1 Cov z , blue,t . 1. (39) ttww w demonstrating that many seeds remain viable for at least 7 12121white,t white,t white,t 2 years. By comparing seedling densities on their elimination As shown in Table 1, the point estimate of this quantity is plots to the maximum densities on control plots, Wright only 0.92, suggesting that blue may not be stably maintained (1978, p. 196) argued that seed density declined by more than in this population. There are, however, two sources of sta- 50% in 6 years. This implies that 1 2 a is less than 0.89, so tistical uncertainty that must be considered: the precision of that a is larger than 0.11. However, this estimate of a is the parameter estimates in Table 1 and the extent to which demonstrably unreliable and systematically biased upward. the observed variation in seed set is representative of the For instance, Wright’s method (1978, table 6.1) implies that long-term variation relevant to this polymorphism. the seed bank was depleted by over 50% within 1 year, but Table 2 addresses the first source of uncertainty with boot- by less than 10% after 4 years. Moreover, his analysis tends strap confidence intervals for E(zt wwhite,t/wblue,t) 2 1, E(zt to overestimate seed bank depletion by comparing the density wblue,t/wwhite,t) 2 1, and E ln{1 1 azt[(wblue,t/wwhite,t) 2 1]}, of plants on the elimination plots, which were chosen because using various values of a. The confidence intervals are based of high initial densities, to the average density on the densest on the bias-corrected and accelerated (BCa)bootstrapmethod square foot of 260 100-ft2 plots. From these data, we con- described in Efron and Tibshirani (1993, ch. 14). The boot- jecture that a is on the order of 0.05 to 0.2, with the lower straps were performed by resampling the multivariate data part of the range being more likely. on seed production, morph frequencies, and plant densities over years. The resulting confidence intervals for both the Protected Polymorphism? necessary, (39), and sufficient, (38), conditions for invasion of blue include values that insure persistence. To check Assuming that blue’s phenotypic dominance implies com- whether both alleles can increase when rare, we constructed plete dominance for fitness, results (26) and (27) imply that bootstrap intervals for the product of the two invasion con- the conditions for protected polymorphism are ditions. In more than 25,000 bootstraps with a 5 0anda 5 ww w 0.2, no examples were found in which neither allele could E z white,t 5 E white,t 1 Cov z , white,t . 1 and (37) ttincrease when rare, thus the product of the invasion condi- 121212wwblue,t blue,t wblue,t tions is positive only when both are positive. Again, the wblue,t confidence intervals include positive values, indicating that E ln 1 1 at 2 1 . 0, (38) []12wwhite,t both invasion conditions can be simultaneously satisfied. FLOWER DIMORPHISM AND FLUCTUATING SELECTION 1293

TABLE 1. Linanthus data from Pearblossom, California (Pearblossom site 1 of Schemske and Bierzychudek 2001).

No. of seeds/plant wblue 1 2 3 4 5 Year Blue White Average wwhite %blue %blueseed Plant density zt 1988 30.45 19.67 21.12 1.55 13.5 19.1 2754 0.269 1989 5.00 3.83 3.99 1.31 13.7 17.0 233 0.004 1991 260.13 302.42 295.71 0.86 15.9 14.0 870 1.190 1992 18.58 13.28 14.06 1.40 14.7 19.1 3268 0.212 1993 2.77 1.12 1.29 2.47 10.4 21.6 1926 0.012 1995 65.62 73.35 72.46 0.89 11.6 10.5 9971 3.341 1998 108.87 125.81 124.29 0.87 9.0 7.9 3432 1.972 Average 70.20 77.07 76.13 1.34 12.7 15.6 3208 1.000 Average RSN 6 1.34 0.85 Geometric mean RSN 1.25 0.80 Weighted average RSN 0.92 1.10 1 This weights the seed numbers by the frequencies reported in the ‘‘% blue’’ column. 2 Frequency estimates for phenotypes are based on complete censuses of blue and white plants in three 50 3 1-m transects associated with the site where seed production was measured (for details, see Schemske and Bierzychudek 2001). 3 This is an approximation for the fraction of seeds carrying a blue allele based on assuming that the parental genotypes follow Hardy-Weinberg frequencies. 4 These densities are based on the censuses used to estimate ﬂower-color frequencies. 5 The relative contributions to the seed bank in each year were estimated by comparing the average number of seeds produced in each year (the product of the average number of seeds per plant and plant density) to the average number across years. 6 Relative seed number, normalized by the number produced by the other morph.

Therefore, the Table 1 data are consistent with the hypothesis would be reduced to about 90% of the observed values. With that fluctuating selection as modeled above maintains the this change, (37) is still satisfied and a protected polymor- flower color polymorphism. The robustness of this conclusion phism would result. will be considered next. Asecondsourceofuncertaintyinouranalysisiswhether One way to quantify how far the data are from meeting the data in Table 1 are representative of the fluctuations ex- the protected polymorphism conditions is to ask what scaling perienced by this population. Over the 11 years of the study, factor would have to be applied to the relative fitnesses of only 7 years had sufficient rain for germination. As noted in blue so that (38) would be satisfied. This can be determined Schemske and Bierzychudek (2001), the combined rainfall by numerically solving in March and April gives the highest correlation with mean seed number. Using March plus April rainfall data from the wblue,t E ln 1 1 ayt 2 1 5 0, (40) nearby Pearblossom weather station, we can test whether the []12wwhite,t seven seed-producing years in Table 1 are typical by com- for y. For 0.05 , a , 0.2, the result is y 5 1.11, that is, an paring them to the 10 years from 1944 to 1957 in which 11% increase in the relative fitness of the blues would allow Epling et al. (1960, fig. 2) observed flowering plants. The them to increase when rare. If the relative fitnesses of the median March plus April rainfall was lower during the flow- blues were increased by this amount, white’s relative fitnesses ering years reported in Epling et al. (1960; 4.19 cm vs. 6.04 cm), but not significantly so (P . 0.5 from a Wilcoxon rank sum test). TABLE 2. Evaluation of the protected polymorphism conditions using Incomplete dominance. How sensitive is our coexistence the data from Table 1. analysis to the assumption that the blue allele has a fully dominant effect on fitness as well as flower color? For sim- 1 Point estimate 95% confidence interval plicity we address this by assuming that heterozygote fitness White: can be expressed as (a) E(ztwwhite,t/wblue,t) 2 1 0.104 (20.091, 0.143) Blue: w12,t 5 hw11,t 1 (1 2 h)wwhite,t (41) (b) E(ztwblue,t/wwhite,t) 2 1 20.076 (20.122, 0.194) (necessary) for some constant h, where w11 denotes the fitness of blue (c) E ln{1 homozygotes. Assuming that the adult genotype frequencies 1 azt[(wblue,t/wwhite,t) 2 1]} do not depart too greatly from Hardy-Weinberg proportions, a 5 0.05 20.004 (20.006, 0.020) the fitness of blue individuals can be approximated by a 5 0.1 20.008 (20.012, 0.041) a 5 0.2 20.016 (20.025, 0.082) wblue,tt5 pw11,tt1 (1 2 p )w12,t, (42) White and blue: (a 3 b) (necessary) 20.008 (20.183, 0.057) where pt is the frequency of the blue allele among flowering sufficient (a 3 c) plants. For a fixed value of h, we can estimate w11,t for each a 5 0.05 20.0004 (20.0009, 0.00004) year by combining (41) and (42), then apply (23) to determine a 5 0.1 20.0008 (20.0018, 0.00007) whether the protected polymorphism conditions are satisfied. a 5 0.2 20.0017 (20.0037, 0.00012) For 0.5 , h , 1, the point estimates indicate that blue still

1 cannot invade, but white can. Equation (40) provides a way BCa bootstrap conﬁdence intervals based on 10,000 bootstrap samples (see Efron and Tibshirani 1993, ch. 14). to quantify the effect of incomplete dominance. We can de- 1294 MICHAEL TURELLI ET AL.

TABLE 3. Approximate quantiles of the long-term frequency variation wblue,t/wwhite,t)presentedinTable1,usingvariousvaluesof of blue-ﬂowered plants based on the stationary distribution of allele a and initial genotype frequencies. The simulations use the frequencies, (34), for various values of the average frequency of the blue allele, E(p), and average annual contribution to the seed bank, a full diploid recursions (18–20), but assume that the seed bank

5 E(at). For each combination of parameters, the upper interval gives begins with Hardy-Weinberg genotype frequencies. The re- the lower and upper quartiles and the lower interval gives the lower sults are summarized in Table 4 in terms of quantiles of the and upper 2.5% quantiles. final frequency of the blue phenotype after seven or 50 seed- producing generations. E(p) (% blue Table 4 illustrates several points. First, as expected from flowers) a 5 0.05 a 5 0.1 a 5 0.2 the fact that the condition for the protection of the blue allele 0.030 (0.059) (0.031, 0.079) (0.019, 0.082) (0.005, 0.080) is not satisfied by our point estimate, the frequency of blue (0.009, 0.150) (0.002, 0.197) (0.00004, 0.275) tends to decrease. However, even with a as high as 0.2, the 0.065 (0.126) (0.087, 0.157) (0.068, 0.167) (0.040, 0.177) buffering effect of the seed bank insures that phenotype fre- (0.042, 0.242) (0.020, 0.300) (0.003, 0.392) 0.100 (0.190) (0.145, 0.228) (0.123, 0.242) (0.087, 0.260) quencies change very slowly. Indeed, even after 50 seed- (0.084, 0.320) (0.050, 0.381) (0.016, 0.477) producing generations, the polymorphism would be maintained. One way to quantify the rate of decline of blue is to calculate an effective relative fitness, denoted we(blue),that termine the factor by which the fitness of blue heterozygotes relates the observed decline in the median allele frequency must be increased relative to whites for blue to invade. As to the dynamics of a fully deterministic model. Separate es- noted above, with h 5 1(bluecompletelydominant),an11% timates of we(blue) can be obtained for each entry in Table 4, relative fitness increase is required. For h 5 0.9, this is re- but all of them equal 0.92 to two significant digits. Finally, duced to 10%; and for h 5 0.8, it is reduced to 9%. Thus, it is notable that the data in Table 1 show a wider range of our qualitative and quantitative results concerning protected fluctuations of phenotype frequencies than shown in Table polymorphism do not depend critically on the assumption of 4, even with a 5 0.2. This suggests sources of variation not complete dominance. captured by our model. Fluctuations in phenotype frequencies. How much frequency variation is expected under our model? Note that the Adequacy of the Model stationary distribution (32) depends on only two parameters, Population geneticists rarely have sufficient data to esti- a 5 E(at)andthecompositeparameterk defined by (30). Thus, once a and the mean allele frequency, E(p), are spec- mate all parameters of a proposed model. Using the estimates ified, the stationary distribution of allele frequencies is com- in Table 1, we can ask whether the model accurately predicts pletely determined. Table 3 gives approximate lower and the changes in morph frequencies reported in Table 1. Despite upper quartiles (25% and 75% quantiles) and 2.5% and 97.5% yearly sample sizes that average 3208, and range from 233 quantiles of the frequency of blue flowers (ignoring depar- to 9971, only two of the frequency estimates between suc- tures from Hardy-Weinberg frequencies) using the stationary cessive samples are significantly different—the change from 2 25 distribution (32) determined as a function of E(p)anda for 14.7% blue in 1992 to 10.4% in 1993 (x1 5 19.5, P 5 10 ) 2 values in the range suggested by the Linanthus data. The and the change from 11.6% in 1995 to 9% in 1998 (x1 5 upper and lower quartiles are comparable to the range of 17.6, P , 1024). (These changes remain highly statistically flower-color frequency variation observed in Table 1 only significant even after correcting for multiple comparisons.) with a 5 0.05 (and E[p] 5 0.065). However, part of the Both years are preceded by statistically significant levels of discordance is attributable to the fact that the seed bank pro- selection (Schemske and Bierzychudek 2001), with blues duces a high autocorrelation in allele frequencies across having higher fitness in 1992 and whites having higher fitness years, so that only a small fraction of the range predicted in in 1995. As expected from (1), the frequency of blue falls Table 3 would be observed over a decade. Rather than con- in 1998 (the first year after 1995 in which flowers were pro- jecturing a joint distribution for the relevant stochastic produced). However, contrary to the prediction of our model, cesses, we have simulated the short-term dynamics of allele the frequency of blue significantly declines from 1992 to frequencies by bootstrapping the seven estimates of (zt, 1993 despite blue’s seed production advantage in 1992. As

TABLE 4. Projected effects of selection based on resampling the empirical estimates of (wblue/wwhite, z)fromTable1.Thevaluesgivenarethe median and lower and upper 2.5% quantiles based on 3000 simulations for the ﬁnal frequency of blue-ﬂowered plants after the indicated number of seed-producing years, starting from various initial frequencies.

Initial frequency a 5 0.05 a 5 0.1 a 5 0.2 7years 0.126 0.123 (0.118, 0.127) 0.120 (0.110, 0.129) 0.114 (0.097, 0.132) 0.190 0.186 (0.179, 0.192) 0.182 (0.169, 0.194) 0.173 (0.147, 0.197) 50 years 0.126 0.106 (0.095, 0.117) 0.089 (0.072, 0.109) 0.061 (0.038, 0.093) 0.190 0.162 (0.146, 0.177) 0.137 (0.111, 0.165) 0.095 (0.062, 0.140) FLOWER DIMORPHISM AND FLUCTUATING SELECTION 1295 indicated by our statistical test of homogeneity, the 1992– Age effects. Epling et al. (1960) found that germination 1993 change in phenotype frequency is not a sampling effect. success declines with seed age, except that 2-year-old seeds If we view the 3268 individuals censused in 1992 and the germinated more readily than 1-year-old seeds. Variation of 1926 individuals censused in 1993 as independent binomial germination rates with age is widely documented (e.g., Phi- samples, the approximate 95% confidence interval for the lippi 1993), and germination delays are expected because new difference in blue frequencies is (20.061, 20.024). What can seeds have a germination inhibitor in their seed coats that is account for this qualitative departure from our predictions? only gradually washed out. Variation in germinability is Our model allows seed viability, germination rates, and known to depend on many factors, including nongenetic ma- seedling viabilities to vary across years but assumes that ternal effects (Baskin and Baskin 1998, ch. 8; Fenner 1985, performance in the seed bank is independent of seed age and ch. 6). Thus, each cohort could have a different age-specific genotype and that seedling viability is independent of ge- germination schedule, depending on the pattern of yearly notype. In reality, seed viabilities and germination rates sure- rainfall. If germination rates are strongly age-dependent, with ly depend on seed age and may depend on: (1) genotype; (2) germination rates peaking after several years, the color com- interactions of genotype with fluctuating conditions; (3) in- position of Epling et al.’s (1960) elimination plots in suc- teractions of genotype with age; and (4) three-way interac- cessive years might reflect morph-specific differences in seed tions involving genotype, age, and conditions. Similarly, production from distinct cohorts. Even without considering seedling viability may vary with genotype and interactions environmental and genetic effects, this generalization—to- between genotype and environmental conditions. We have ward the full age-structured model considered by Templeton no new data to test any of these possibilities, but reanalysis and Levin (1979)—would require estimates of age-specific of the data reported in Epling et al. (1960) suggest significant germination rates that are not available for Linanthus. age effects and/or fluctuating selection prior to seed set. Migration. Another possibility is that the frequency dif- Unexplained variation in preseed-set performance. In their ferences reflect rare long-distance dispersal events (mediated figure 10B, Epling et al. (1960) report flower-color frequencies perhaps by whirlwinds that are common in the summer or from 1948 to 1954 observed on three polymorphic 100-ft2 flash floods) from regions with different genetic composi- elimination plots, separated by 50 ft, from which all flowering tions. This might be plausible if there were populations near- plants were removed before seed set each year. There are two by with quite different phenotype frequencies. However, the noteworthy observations in addition to the demonstration of mixed composition at this site is typical of the region. The seed bank longevity. First, there are no statistically significant nearest population with a high frequency of blues is 1.5 km differences between the phenotype frequencies observed in away (however, in the same direction, there is a population 1948 and 1954 on any of the three plots. In two plots the that is monomorphic for white only 0.5 km away). Although observed frequency of blue went up insignificantly, but in the some gene flow may occur at such distances, it could not third it went down. Thus, as emphasized by Wright (1978, p. significantly alter allele frequencies unless local effective 197), there is no evidence for consistent directional selection population sizes remained extremely small. This hypothesis on seed viability. Second, contrary to Wright’s assertion that is inconsistent with the observed long-term stability of phe- there were no significant changes in the color frequencies over notype frequencies found by Epling et al. (1960) and Schem- the 5 years, two of the three plots—the ones with the largest ske and Bierzychudek (2001) and estimates of Nm from al- sample sizes—show statistically heterogeneous phenotype fre- lozyme frequencies (D. W. Schemske and P. Bierzychudek, 22 quencies (the three samples yield, xx345 5.2, P . 0.2; 5 unpubl. data). Wright (1978, p. 222) also concluded that mi- 2 23.1, P , 0.001; and x4 5 14.5, P , 0.01, respectively). The gration could not explain the observed frequency changes. two sites with variable frequencies are separated by only 50 ft; and for both, the significant heterogeneity is caused by 9% DISCUSSION frequency changes between 1952 and 1953. Surprisingly, the frequency of blue significantly increased at one site, but de- We have attempted to understand the white versus blue creased at the other. This suggests fine-scale spatial variation flower-color dimorphism in a population of L. parryae in in the composition of the seed bank and/or the age- and/or terms of a simple model of natural selection with overlapping genotype-dependent germination propensities of the seeds. To generations that captures the fluctuating absolute and relative explain these data solely in terms of genotype-dependent ger- fitnesses documented in Schemske and Bierzychudek (2001). mination differences, the relative germination rate for blue We presented a series of theoretical analyses leading up to would have to be 1.45 (0.69) to explain the 9% increase (de- adiploidmodelwithdominance.Ourtheoreticaltreatment crease). Similarly, the 1992–1993 frequency data in Table 1 follows a long tradition of analyzing fluctuating selection and can be explained by a relative germination rate for blue of its potential to maintain genetic variation (reviewed by Fel- 0.67. This hypothetical selection is quite intense, but com- senstein 1976; Hedrick 1986; Gillespie 1991). We have pro- parable to the fecundity selection documented in Table 1. In- vided analytical conditions for protected polymorphism and terestingly, this scenario requires that in 1993 when the blues used diffusion approximations to describe the long-term had a significant seed set advantage, they had a significant properties of these polymorphisms. However, the data doc- germination rate (or viability) disadvantage of comparable umenting intense fluctuating selection and a long-lived seed magnitude. Alternatively, the unexplained variation may re- bank raise novel questions about both using weak-selection flect age-dependent germination rates and fluctuations in the diffusion approximations (elaborated in the Appendix) and genotype frequencies of newly produced seeds, as shown by applying long-term predictions derived from the diffusion. the column labeled ‘‘% blue seed’’ in Table 1. We have shown that the fluctuating fitnesses estimated 1296 MICHAEL TURELLI ET AL. from nature are statistically consistent with maintaining the Chesson, P. L., and R. R. Warner. 1981. Environmental variability observed polymorphism. This conclusion relies critically on promotes coexistence in lottery competitive systems. Am. Nat. 117:923–943. both the existence of a persistent seed bank and significant Christiansen, F. B. 1988. The Wahlund effect with overlapping variation in yearly seed set. Our analysis makes clear why generations. Am. Nat. 131:149–156. the white-flowered morph is more common than the blue Dempster, E. R. 1955. Maintenance of genetic heterogeneity. Cold despite the fact that both the arithmetic and geometric means Spring Harb. Symp. Quant. Biol. 20:25–32. Efron, B., and R. J. Tibshirani. 1993. An introduction to the boot- of white’s relative fitnesses are less than one. The key ob- strap. Chapman and Hall, New York. servation is that the white morph produces consistently more Ellner, S., and N. G. Hairston Jr. 1994. Role of overlapping gen- seeds than the blues in the years with the heaviest rainfall erations in maintaining variation in a fluctuating environment. and highest seed set. These years contribute most to the seed Am. Nat. 143:403–417. bank and thus to subsequent generations. This provides a Ellner, S., and A. Sasaki. 1996. Patterns of genetic polymorphism maintained by fluctuating selection with overlapping genera- simple genetic analog of the species-diversity maintaining tions. Theor. Popul. Biol. 50:31–65. storage effect proposed by Chesson and Warner (1981; for Ellner, S. P., N. G. Hairston Jr., C. M. Kearns, and D. Babai. 1999. apolygenicexample,seeEllnerandHairston1994). The roles of fluctuating selection and long-term diapause in mi- Despite the extreme fluctuations in the observed relative croevolution of diapause timing in a freshwater copepod. Evo- lution 53:111–122. fitnesses, the morph frequencies have changed relatively lit- Epling, C., and T. Dobzhansky. 1942. Genetics of natural popu- tle. This reflects the strong damping effect of the seed bank lations. VI. Microgeographic races in Linanthus parryae. Ge- (cf. Gottlieb 1974). This damping also implies that even a netics 27:317–332. study that spans more than a decade may be insufficient for Epling, C., H. Lewis, and F. M. Ball. 1960. The breeding group studying the range of environmental conditions responsible and seed storage: a study in population dynamics. Evolution 14: 238–255. for the polymorphism or for determining whether the poly- Felsenstein, J. 1976. The theoretical population genetics of variable morphism is transient or stable. As shown by the numerical selection and migration. Annu. Rev. Genet. 10:253–280. results in Tables 3 and 4, the long-term fluctuations predicted Fenner, M. 1985. Seed ecology. Chapman and Hall, London. by the diffusion’s stationary distribution would take well over Gillespie, J. H. 1973. Natural selection with varying selection coefficients: a haploid model. Genet. Res. 21:115–120. 50 years to observe with plausible levels of seed longevity. ———. 1974. Polymorphism in patchy environments. Am. Nat. Equally sobering is the fact that thorough studies of adult 108:145–151. fitness spanning more than a decade would have to be sup- ———. 1991. The causes of molecular evolution. Oxford Univ. plemented by even more elaborate and sustained studies of Press, Oxford, U.K. seed bank demography and selection to understand all of the Gottlieb, L. D. 1974. Genetic stability in a peripheral isolate of Stephanomeria exigua ssp. coronaria that fluctuates in population biology relevant to the transient behavior of this ‘‘simple’’ size. Genetics 76:551–556. polymorphism. As demonstrated by the insights that have Grant, B. R., and P. R. Grant. 1989. Evolutionary dynamics of a emerged from the decades of study of Darwin’s finches (Grant natural population. Univ. of Chicago Press, Chicago, IL. 1986; Grant and Grant 1989; Grant and Grant 1997), progress Grant, P. R. 1986. Ecology and evolution of Darwin’s finches. Princeton Univ. Press, Princeton, NJ. in merging evolutionary genetics with ecology may often Grant, P. R., and B. R. Grant. 1997. Genetics and the origin of bird require such long-term studies. species. Proc. Natl. Acad. Sci. USA 94:7768–7775. Haldane, J. B. S., and S. D. Jayakar. 1963. Polymorphism due to ACKNOWLEDGMENTS selection of varying direction. J. Genetics 58:237–242. Hatfield, J. S., and P. L. Chesson. 1989. Diffusion analysis and We thank N. H. Barton, P. Chesson, J. H. Gillespie, L. D. stationary distribution of the two-species lottery competition Gottlieb, R. Haygood, H. A. Orr, S. H. Orzack, and two model. Theor. Popul. Biol. 36:251–266. anonymous reviewers for helpful comments and discussion. Hedrick, P. W. 1986. Genetic polymorphism in heterogeneous environments: a decade later. Annu. Rev. Ecol. Syst. 17:535–566. Financial support for DWS and PB’s fieldwork was provided ———. 1995. Genetic polymorphism in a temporally varying en- by National Science Foundation BSR-8918246 to the Uni- vironment: effects of delayed germination or diapause. Heredity versity of Washington (DWS) and National Science Foun- 75:164–170. dation RUI BSR-8919674 to Pomona College (PB). Other Karlin, S., and U. Liberman. 1975. Random temporal variation in support was provided by Pomona College Schenck Funds selection intensities: one-locus, two-allele model. J. Math. Biol. 2:1–17. and a Pomona College Seaver Grant to PB. Additional sup- Karlin, S., and H. M. Taylor. 1981. A second course in stochastic port was provided by National Science Foundation grant DEB processes. Academic Press, New York. 9527808 (MT). Kimura, M. 1954. Process leading to quasi-fixation of genes in natural populations due to random fluctuation of selection intensities. Genetics 39:280–295. LITERATURE CITED Levikson, B., and S. Karlin. 1975. Random temporal variation in Abramowitz, M., and I. A. Stegun. 1964. Handbook of mathematical selection intensities acting on infinite diploid populations: dif- functions. National Bureau of Standards, Washington, DC. fusion method analysis. Theor. Popul. Biol. 8:292–300. Baskin, C. C., and J. M. Baskin. 1998. Seeds. Academic Press, San Orzack, S. H. 1993. Life history evolution and population dynamics Diego, CA. in variable environments: some insights from stochastic de- Bodmer, W. M. 1965. Differential fertility in population genetics mography. Pp. 63–104 in J. Yoshimura and C. W. Clark, eds. models. Genetics 51:411–424. Adaptation in stochastic environments. Springer-Verlag, New Chesson, P. L. 1985. Coexistence of competitors in spatially and York. temporally varying environments: a look at the combined effects Philippi, T. 1993. Bet-hedging germination of desert annuals: be- of different sorts of variability. Theor. Popul. Biol. 28:263–287. yond the first year. Am. Nat. 142:474–487. ———. 1994. Multispecies competition in variable environments. Prout, T. 1968. Sufficient conditions for multiple niche polymor- Theor. Popul. Biol. 45:227–276. phism. Am. Nat. 102:493–496. FLOWER DIMORPHISM AND FLUCTUATING SELECTION 1297

Seger, J., and H. J. Brockmann. 1987. What is bet-hedging? Oxf. p (1 2 p )(2 w 2 w ) D p 5 tt1,t 2,t . (A6) Surv. Evol. Biol. 4:182–211. st p (2 2 p )w 1 (1 2 p )2w Schemske, D. W., and P. Bierzychudek. 2001. Evolution of flower tt1,tt2,t color in the desert annual Linanthus parryae: Wright revisited. The selection coefficient st is again defined by (A1), and the analog Evolution 55:1269–1282. of (A5) is Subramaniam, B., and M. D. Rausher. 2000. Balancing selection Dp 5 a(1 1 y )p (1 2 p )222s {1 2 s [1 2 (1 2 p )]1 O(s )}. in a floral polymorphism. Evolution 54:691–695. ttttttt Templeton, A. R., and D. A. Levin. 1979. Evolutionary conse- (A7) quences of seed pools. Am. Nat. 114:232–249. Consistency Criterion for Identifying the Constants in the Tuljapurkar, S. 1990. Delayed reproduction and fitness in variable Infinitesimal Moments environments. Proc. Natl. Acad. Sci. USA 87:1139–1143. Turelli, M. 1977. Random environments and stochastic calculus. As described in the text, the interpretation of the constants that Theor. Popul. Biol. 12:140–178. appear in the infinitesimal moments is ambiguous. This is most ———. 1981. Temporally varying selection on multiple alleles: a easily demonstrated by example. Note, for instance, that under the diffusion analysis. J. Math. Biol. 13:115–129. assumptions (A3), Wolfram, S. 1996. Mathematica. 3rd ed. Cambridge Univ. Press, E(sytt) 5 Cov(s t, y t) 5 Cov[ln(1 1 s t), y t] 1 o(e). (A8) Cambridge, U.K. Wright, S. 1943a. Isolation by distance. Genetics 28:114–138. Because the infinitesimal moments involve only the leading-order ———. 1943b. An analysis of local variability of flower color in terms, the constant C in the infinitesimal mean (13a) might plausibly Linanthus parryae. Genetics 28:139–156. be interpreted as either Cov(w1,t/w2,t, zt)orCov[ln(w1,t/w2,t), zt]. In ———. 1948. On the roles of directed and random changes in gene the limit as e → 0, this choice makes no difference. However, what frequency in the genetics of populations. Evolution 2:279–294. we want to understand is not the limiting case, but actual data whose ———. 1978. Evolution and the genetics of populations. Vol. 4 dynamics are described by the discrete-time stochastic process with Variability within and among natural populations. Univ. of Chi- appreciable selection and appreciable fluctuations. For the data in cago Press, Chicago, IL. Table 1, Cov(w1,t/w2,t, zt) 520.50, whereas Cov[ln(w1,t/w2,t), zt] 5 20.38, so the interpretation of C can significantly affect our pre- s2 2 2 Corresponding Editor: O. Savolainen dictions. Similarly, might be interpreted as E{[(w1,t/w2,t) 1] }, Var(w1,t/w2,t), or Var[ln(w1,t/w2,t)], because all three differ only by terms that are o(e)accordingto(A3). We propose the following consistency criterion for identifying APPENDIX the constant: The diffusion approximation should be the same Derivation of the Infinitesimal Mean and Variance whether selection is quantified by To derive the infinitesimal moments for the haploid model de- w1,t scribed by (1) and (2), we define the selection coefficient st by 5 1 1 st or (A9a) w2,t w 1,t 5 1 1 s ; (A1) t w2,t w2,t 5 1 1 s˜ . (A9b) w t and we describe the relative fluctuations in the contributions to the 1,t seed bank by We will first show that both parameterizations lead to the same diffusion when assumptions (A3) are met, then show how these att[ az [ a(1 1 y t), (A2) calculations lead to a specific choice for the constants. Define m, 2 where a 5 E(at)andE(yt) 5 0. Following the standard diffusion- s ,andC as in (A3). We can make analogous assumptions con- approximation conventions (Karlin and Taylor 1981, ch. 15, es- cerning s˜t, and define pecially pp. 184–188), we assume that for some small value e,the 22 E(s˜tt) 5m˜ 5 O(e), E(s˜ ) 5s˜ 5 O(e), (A10a) moments of the random variables st and yt satisfy 22 22 E(yta) 5 CV 5 O(e), E(stt) 5m5O(e), E(s ) 5s 5O(e), (A3a) ˜ 22 E(s˜tty ) 5 C 5 O(e), and (A10b) E(yta) 5 CV 5 O(e), E(sy tt) 5 C 5 O(e), and (A3b) i j i j E(s˜t yt ) 5 o(e) for i 1 j $ 3. (A10c) E(syt t ) 5 o(e) for i 1 j $ 3, (A3c) The expansion analogous to (A5) is where O(e)denotesaquantitythatapproacheszeroatthesamerate 3 as e as e → 0, and o(e)denotesasmallerquantitythatapproaches Dptttttttt52a(1 1 y )p (1 2 p )[s˜ (1 2 s˜ 1 ps˜ ) 1 O(s˜ )]. (A11) zero faster than e.(Thecompletederivationalsorequiresscaling The corresponding infinitesimal moments are the generation time; see Turelli 1977; Karlin and Taylor 1981, ch. 15.) 1 Using (1), the allele frequency dynamics follow m˜ (p) 5 ap(1 2 p) g˜ 1 C˜ 1s˜ 222s˜ p and (A12a) 122 Dp 5 a (p9 2 p ) [ a D p, (A4) ttt tts t v˜(p) 5 a22s˜ [ p(1 2 p)] 2 , (A12b) where Dspt 52pt9 pt describes the effects of selection in generation 2 t. The diffusion assumptions (A3) allow us to approximate the dy- with g˜ 52m˜ Ωs˜ .Toreconcilethealternativeexpressionsforthe infinitesimal moments, note that namics by expanding Dpt in a Taylor series around st 5 0andyt 5 0andretainingonlythesecond-andlower-orderterms.Ex- w 1 2,t 5 1 1 s˜ 551 2 s 1 s231 O(s ). (A13) panding the haploid expression for Dspt in powers of st, we see that ttt w1,tt1 1 s 3 Dptttttt5 a(1 1 y )[p (1 2 p )s (1 2 s ) 1 O(s )]. (A5) Thus, This expansion and assumptions (A3) produce the expressions given m˜ 52m1s2, (A14a) in (13) for m(p) 5 E(Dpt z pt 5 p)andv(p) 5 Var(Dpt z pt 5 p). The corresponding diffusion for diploid selection with dominance s˜ 225s , (A14b) is derived in the same way, except that instead of using (2) to C˜ 52C, and (A14c) describe selection in (A4), we approximate the effect of selection by assuming that seedlings follow Hardy-Weinberg genotype pro- g˜ 52g. (A14d) portions so that Using these, it follows that 1298 MICHAEL TURELLI ET AL.

m(p) 5 m˜ (p) and (A15a) g˜ 52g, (A17a) v(p) 5 v˜(p). (A15b) s˜ 225s , and (A17b) Hence, in the weak-selection, small-ﬂuctuation limit, the alternative C˜ 52C. (A17c) parameterizations of selection, (A9), produce identical results. This leads to the choices This does not tell us what to do with our data, however. As noted above, (A3) and (A10) seem compatible with setting ww1,t 1,t g5E ln , s52 Var ln , ww m5E(w /w ) 2 1, s52 Var(w /w ) []122,t [] 122,t 1,t 2,t 1,t 2,t

C 5 Cov(w1,t /w2,tt, z ), or (A16a) w1,t C 5 Cov zt, ln and (A18a) 2 12w2,t m˜ 5 E(w2,t /w1,t) 2 1, s˜ 5 Var(w2,t /w1,t), [] ˜ C 5 Cov(w2,t /w1,tt, z ). (A16b) ww g˜ 5 E ln2,t , s˜ 2 5 Var ln2,t , However, these moments will not generally satisfy (A14), so the []12ww1,t [] 121,t resulting infinitesimal means and variances (and consequently the predictions for E[p]andSD[p]) will differ depending on whether w C˜ 5 Cov z , ln2,t , (A18b) (A16a) or (A16b) are used. Our consistency criterion suggests that t w the coefficients should be defined so that []121,t discussed in the text (see eq. 14).